The tension in a suspension cable is T = √(H² + V²) with H = wL² / 8d, so it climbs as the sag falls: on a 30 m span carrying 0.75 kN/m per cable, a 3 m sag gives 30.3 kN and a 1.5 m sag gives 57.4 kN. At a safety factor of 5 and a cable strength of 40-60 kg per mm², 30.3 kN needs a cable of 18-22 mm diameter, so a fixed 12-16 mm cable only suits shorter spans of about 9-23 m. Check the end terminations and the anchors first, because they are weaker than the cable.
Part of Bridge Types & Construction
A pedestrian suspension bridge with two main steel cables, vertical hangers and a plank deck, with the cable, termination and anchor forces worked out from statics.
The Cable Carries Its Load As A Parabola, And Sag Sets The Tension
A cable carrying a load spread evenly along its horizontal span hangs in a parabola. Cut it at midspan and the two halves are in balance with a horizontal pull H, the same at every point along the cable, and a vertical component that grows from zero at midspan to V = wL / 2 at each support. These are the standard results for a parabolic cable, found in any structural-mechanics text such as R. C. Hibbeler’s Structural Analysis; I am citing that from memory and have not rechecked it here. The parent article states the qualitative rule: more droop means less tension on the cables and anchors, but more vertical distance to climb at the ends.
In numbers: horizontal pull H = w L² / (8 d), vertical end force V = w L / 2, and the tension at the support T = √(H² + V²), where w is the load per metre of span on that cable, L the span and d the sag at midspan. The cable also gets longer with sag: length ≈ L + 8 d² / (3 L).
The parent says to size the cable by breaking strength: a steel cable breaks at roughly 40-60 kg per mm² of cross-sectional area, 12-16 mm cable is typically sufficient for a pedestrian bridge, and cart traffic needs engineering advice. The arithmetic below shows what “typically sufficient” depends on.
What You Need
- Two main cables of known diameter and condition. Salvaged cable should be inspected for broken wires, kinks and rust.
- Hangers at 1-1.5 m, per the parent.
- Cross-beams and planks.
- Terminations — thimbles with clips, spliced eyes, or sockets.
- Anchors on both banks. See Abutment Construction.
- A survey of span and level difference. See Surveying Techniques.
Method: Size A Cable Bridge From The Loads
Step 1: Collect the loads Add the dead load of deck, cables, hangers and rails per metre of span to the live load per metre. Split between the two cables.
Step 2: Choose a sag Pick d as a fraction of the span. Compute H and T for two or three values and compare.
Step 3: Pick the cable Required breaking strength = safety factor × T, adjusted for the termination efficiency. Convert to area using the cable strength per mm², then to diameter.
Step 4: Size the anchors The anchor must resist H and also the vertical component V. See the example.
Step 5: Check the hangers Each hanger carries the load of its panel. At 1.25 m spacing and 0.75 kN/m per cable, that is 0.75 × 1.25 = 0.94 kN, so a hanger must break at 5 × 0.94 = 4.7 kN.
Worked Example: A 30 m Footbridge
Assumptions, which are mine and not the parent’s: the deck is 1.5 m wide, of 5 cm planks at 500 kg/m³ (seasoned softwood, 400-500), so 1.5 × 0.05 × 500 = 37.5 kg/m = 0.37 kN/m; cables, hangers and rails add 0.15 kN/m, giving a dead load of 0.52 kN/m in total. Live load: one person of 100 kg every metre, 100 × 9.81 / 1000 = 0.98 kN/m. Total 1.50 kN/m, shared by two cables: w = 0.75 kN/m each. Safety factor 5 from the parent. Cable strength 40-60 kg per mm² of the whole circular cross-section, so 392-588 N/mm² (multiply kgf by 9.81).
Tension against sag, L = 30 m.
| Sag d | Sag ratio | H = wL² / 8d | V = wL / 2 | T |
|---|---|---|---|---|
| 1.5 m | 1/20 | 56.3 kN | 11.3 kN | 57.4 kN |
| 2 m | 1/15 | 42.2 kN | 11.3 kN | 43.7 kN |
| 3 m | 1/10 | 28.1 kN | 11.3 kN | 30.3 kN |
| 4 m | 1/7.5 | 21.1 kN | 11.3 kN | 23.9 kN |
| 6 m | 1/5 | 14.1 kN | 11.3 kN | 18.0 kN |
For d = 3 m: H = 0.75 × 30² / (8 × 3) = 675 / 24 = 28.1 kN; V = 0.75 × 30 / 2 = 11.25 kN; T = √(28.1² + 11.25²) = 30.3 kN.
Cable for d = 3 m.
- Required breaking strength: 5 × 30.3 = 151.5 kN, which is 151,500 / 9.81 = 15,400 kgf.
- Area at 60 kg/mm²: 15,400 / 60 = 257 mm²; at 40 kg/mm²: 385 mm².
- Diameter = √(4A / π): 18.1 mm to 22.1 mm, so a cable of 18-22 mm.
What 12 and 16 mm can span, at a sag of one-tenth of the span. For d = L / 10, H = 1.25 w L, V = 0.5 w L and T = 1.346 w L. A 12 mm cable has an area of 113 mm² and breaks at 4,520-6,790 kgf (44-67 kN); working tension at a factor of 5 is 8.9-13.3 kN, so L = 8.9 / (1.346 × 0.75) to 13.3 / (1.346 × 0.75) = 8.8 m to 13.2 m. A 16 mm cable has an area of 201 mm², breaks at 79-118 kN, working tension 15.8-23.7 kN, so L = 15.6 m to 23.4 m. Together that is about 9-23 m under this loading. The parent’s “typically sufficient” is true for spans of that order and false at 30 m.
Terminations. Wire-rope clips correctly applied typically give about 80% of the rope strength; this is the general manufacturer figure for clip terminations (Crosby and others), recalled and not checked here. The cable must then break at 151.5 / 0.80 = 189 kN (19.3 tf), so at 40-60 kg/mm² the area is 322-483 mm² and the diameter 20.2-24.8 mm.
Anchors. Per cable at d = 3 m: H = 28.1 kN pulls toward the span, and the cable arrives at the anchor angled arctan(11.25 / 28.1) = 21.8 degrees above horizontal, so V = 11.3 kN tries to lift the anchor. For a gravity block resting on the ground with a friction coefficient of 0.5 (an assumption) and a sliding factor of 1.5, count friction only and ignore passive soil resistance: 0.5 × (W − 11.3) ≥ 1.5 × 28.1, so W ≥ 84.3 + 11.3 = 95.7 kN, about 9.8 tonnes, or 9.75 / 2.6 = 3.75 m³ of solid stone at 2,600 kg/m³. That is per cable at each end, four blocks in all. Note that Suspension Bridge says a gravity block must weigh at least 1.5 T, which here is 45.5 kN. By friction alone 0.5 × (45.5 − 11.3) = 17.1 kN against a pull of 28.1 kN is a sliding factor of 0.61 and it would slide. That article also counts passive soil resistance, which I have not. Anchor into rock or sound ground, and load-test an anchor before rigging the cable.
Common Mistakes
- Using the diameter without the span — the answer depends on span, sag and load.
- Forgetting the termination — the cable is rarely what breaks.
- Setting a tight sag to remove the bounce — halving the sag from 3 m to 1.5 m nearly doubles the cable force, from 30.3 to 57.4 kN.
- Ignoring dynamic effects — a crowd walking in step makes a flexible deck sway. Keep the live load assumption honest and require break-step crossing.
- Counting a gravity block as a rock anchor — its sliding resistance is friction, weight and the soil in front of it.
Safety
Stored energy and falls
A cable under tens of kilonewtons of tension snaps back along its line with great force when it breaks; keep everyone out of line with the cable while it is tensioned and when it is tested. A failure of an anchor or termination drops the whole deck into the gorge. Test with sandbags from the bank, not with people on the span. Limit the crowd to the load the calculation used: here, 1 person per metre of span.
What’s Next
- Learn next: Dead Load vs Live Load — where the 0.98 kN/m came from and what real crowds weigh
- Combine with: Foundations & Abutments — anchoring against the pull